Subscribe to this blog

Follow by Email

Search This Blog

The Five Numbers Game

Q: You are in a game where numbers from 1 to 5 are drawn in a random order (without repetition). You win the game if you do not get 3 numbers in successive order (not necessarily consecutive) that are increasing. What is the probability that you will win this game?

A: Problems of this particular nature are trivially solved by using/remembering that the number of permutations that avoid an increasing triplet of 3 numbers are simply the Catalan number for n=5.
Catalan number for a given \(n\) is
$$
C_n = \frac{1}{n+1}\binom{2n}{n}
$$
Plugging in \(n = 5\) into this yields
$$
C_5 = \frac{1}{6}\frac{10!}{5!5!} = 42
$$
The total number of ways that 5 numbers can be drawn is \(5! = 120\). The probability that you will win the game is \( 1 - \frac{42}{120} = 65\%\).

If you are looking to buy some books in probability here are some of the best books to learn the art of Probability

Discovering Statistics Using R
This is a good book if you are new to statistics & probability while simultaneously getting started with a programming language. The book supports R and is written in a casual humorous way making it an easy read. Great for beginners. Some of the data on the companion website could be missing.

A Course in Probability Theory, Third Edition
Covered in this book are the central limit theorem and other graduate topics in probability. You will need to brush up on some mathematics before you dive in but most of that can be done online

Discovering Statistics Using R
This is a good book if you are new to statistics & probability while simultaneously getting started with a programming language. The book supports R and is written in a casual humorous way making it an easy read. Great for beginners. Some of the data on the companion website could be missing.

Linear Algebra (Dover Books on Mathematics)
An excellent book to own if you are looking to get into, or want to understand linear algebra. Please keep in mind that you need to have some basic mathematical background before you can use this book.

Linear Algebra Done Right (Undergraduate Texts in Mathematics)
A great book that exposes the method of proof as it used in Linear Algebra. This book is not for the beginner though. You do need some prior knowledge of the basics at least. It would be a good add-on to an existing course you are doing in Linear Algebra.

Follow @ProbabilityPuzIf you are looking to learn time series analysis, the following are some of the best books in time series analysis.

Introductory Time Series with R (Use R!)
This is good book to get one started on time series. A nice aspect of this book is that it has examples in R and some of the data is part of standard R packages which makes good introductory material for learning the R language too. That said this is not exactly a graduate level book, and some of the data links in the book may not be valid.

Econometrics
A great book if you are in an economics stream or want to get into it. The nice thing in the book is it tries to bring out a oneness in all the methods used. Econ majors need to be up-to speed on the grounding mathematics for time series analysis to use this book. Outside of those prerequisites, this is one of the best books on econometrics and time series analysis.